New Zealand’s Devon Conway made exactly 200 on his Test debut before England captain Joe Root helped the hosts recover from a top-order collapse on the second day of the series opener at Lord’s on Thursday.
Opening batsman Conway was last man out in a first-innings total of 378.
England, who slumped to 18-2, ended the day on 111-2, a deficit of 267 runs.
Burns was 59 not out and Root unbeaten on 42, with their partnership worth 93 runs.
“I think it’s reasonably evenly poised,” New Zealand paceman Tim Southee, who took 1-17 in 11 overs, told reporters after stumps.
“There’s a little bit there for the bowlers.”
Southee, turning to Conway’s remarkable contribution, added: “It was an incredible innings. He’s a special player, we’ve known that for a little while now.
“He works extremely hard, he’s always willing to learn and continue to ask questions. Everyone’s just extremely proud and pleased for him.”
New Zealand, who resumed on 246-3, looked like they might bat England out of the game while Conway and fellow left-hander Henry Nicholls (61) were sharing a fourth-wicket stand of 174.
But they lost four wickets for just six runs as they slumped from 288-3 to 294-7, with Mark Wood, the fastest member of England’s pace quartet, instigating a collapse when he had Nicholls caught at long leg by Test debutant Ollie Robinson.
Sussex paceman Robinson led England’s attack with 4-75 in 28 overs.
It was a fine effort by Robinson, who apologised following stumps on Wednesday after racist and sexist tweets he had posted as a teenager were published again.
“Yesterday was a hard day, it tested his personality and character,” said England batting coach Graham Thorpe of Robinson.
“We have to support him as a player in our team we were really happy with the way he performed today.”
Assessing the match situation, the former England batsman added: “From where we were this morning, we’re really happy to fight back.
“We said yesterday that New Zealand didn’t get away from us, we didn’t bowl badly so we had to make inroads and 378, we’d have taken that from the position that we were in.”
Earlier, South Africa-born left-hander Conway was in sight of becoming the very first batsman in the 144-year history of Test cricket to carry his bat throughout the entire innings in their first knock in the format when he was run out to end a stay of more than nine-and-a-half hours.
Nevertheless, it needed a review of a close call before it was confirmed Root had taken the bails off in time.
Even so, Conway was only the seventh batsman to make a double hundred on Test debut.
He faced 347 balls, with 22 fours and went to 200 in style when he hooked Wood for a brilliant six.
New Zealand No 11 Neil Wagner, also born in South Africa, made 25 not out that included a superb straight six off Stuart Broad.
New Zealand, who after this two-match series face India in the inaugural World Test Championship final at Southampton later this month, struck with the new ball despite being without left-arm spearhead Trent Boult, who had been on family leave following a stint in the Indian Premier League.
Towering paceman Kyle Jamieson had Dom Sibley lbw for a duck on ‘umpire’s call’ after a review.
But there was no doubt when Zak Crawley was caught behind off Southee for two.
Burns and Root, however, then held firm in an often attritional stand, although left-hander Burns did go to fifty in 90 balls.